Using collisions to control movements of objects in an animation
Yes - this is precisely what Rigid Body Physics is for.
Your objects will need to be 3-dimensional (your example image looks to only be 2D) and set up with Rigid Body on each. The green one you have have set as 'Passive' and 'Animated' in its settings (so that you can animate its position) and the violet one as 'Active'. You would need to setup a ground plane (also as 'Passive') for them to rest upon (or they will just fall under the influence of gravity) or add a Rigid Body Constraint to restrict its movement.
Running the simulation and moving the green object should result in the violet one responding and being pushed out of the way.